Problem #PRU-58107

Problems Geometry Plane geometry Area Finding the area by rearrangment Set theory and logic Set theory Inclusion-exclusion principle Methods Pigeonhole principle Pigeonhole principle (area and volume) Proof by contradiction
 14–15  3.0

Problem

a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.

b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than \(\frac{1}{9}\).